23.1Properties of Electric Charges23.2 Insulators and Conductors23.3Coulomb’s Law23.4 The Electric Field23.5 Electric Field of a Continuous Charge Distribution23.6 Electric Field Lines23.7 Motion of Charged Particles in a Uniform Electric Field

 

PHY102: CHAPTERS 23 – 26LECTURER: DR M.A. ELERUJA

23.0  Introduction.

• The electromagnetic force between charged particles is one of the fundamental forces of nature. 

• For us to understand the nature of this force we need to describe some of the basic properties of electric forces, followed by the discussion on Coulomb’s law, which is the fundamental law governing the force between any two charged particles. 

• Next, we introduce the concept of an electric field associated with a charge distribution and describe its effect on other charged particles. 

• We then show how to use Coulomb’s law to calculate the electric field for a given charge distribution.

•  We conclude the chapter with a discussion of the motion of a charged particle in a uniform electric field.

23.1 Some properties of Electric charges.

Some simple experiments demonstrate the existence of electric forces and charges.

These include: 1. After running a comb through your hair on a dry day, you will find that the comb attracts bits of paper. 2. Similar effect occurs when materials such as glass or rubber are rubbed with silk or fur.3. When an inflated balloon is rubbed with wool, the balloon adheres to a wall, often for hours.When materials behave in this way, they are said to be electrified, or to have become electrically charged.

In a series of simple experiments, it is found that there are two kinds of electric charges, which were given the names positive and negative by Benjamin Franklin (1706–1790).

To verify that there are two kinds of charges, consider a hard rubber rod that has been rubbed with fur and then suspended by a nonmetallic thread, as shown in Figure 23.1. When a glass rod that has been rubbed with silk is brought near the rubber rod, the two attract each other (Fig. 23.1a).

On the other hand, if two charged rubber rods (or two charged glass rods) are brought near each other, as shown in Figure 23.1b, the two repel each other.

This observation shows that the rubber and glass are in two different states of electrification. On the basis of these observations, we conclude that like charges repel one another and unlike charges attract one another.

Using the convention suggested by Franklin:

electric charge on the glass rod is called positive 

and that on the rubber rod is called negative.

Therefore, any charged object attracted to a charged rubber rod (or repelled by a charged glass rod) must have a positive charge, and any charged object repelled by a charged rubber rod (or attracted to a charged glass rod) must have a negative charge.

Another important aspect of Franklin’s model of electricity is the implication that electric charge is always conserved.

That is, when one object is rubbed against another, charge is not created in the process.

The electrified state is due to a transfer of charge from one object to the other.

One object gains some amount of negative charge while the other gains an equal amount of positive charge.For example, when a glass rod is rubbed with silk, the silk obtains a negative charge that is equal in magnitude to the positive charge on the glass rod.

From our understanding of atomic structure we now know that negatively charged electrons are transferred from the glass to the silk in the rubbing process. 

Similarly, when rubber is rubbed with fur, electrons are transferred from the fur to the rubber, giving the rubber a net negative charge and the fur a net positive charge.

This process is consistent with the fact that neutral, uncharged matter contains as many positive charges (protons within atomic nuclei) as negative charges (electrons).

Robert Millikan (1868–1953) discovered that electric charge always occurs as some integral multiple of a fundamental amount of charge e. In modern terms, the electric charge q is said to be quantized, where q is the standard symbol used for charge.

23.2 Insulators and Conductors

It is convenient to classify substances in terms of their ability to conduct electric charge:Electrical conductors are materials in which electric charges move freely, whereas electrical insulators are materials in which electric charges cannot move freely.

Materials such as glass, rubber, and wood fall into the category of electrical insulators.When such materials are charged by rubbing, only the area rubbed becomes charged, and the charge is unable to move to other regions of the material.

In contrast, materials such as copper, aluminum, and silver are good electrical conductors. When such materials are charged in some small region, the charge readily distributes itself over the entire surface of the material.

If you hold a copper rod in your hand and rub it with wool or fur, it will not attract a small piece of paper. However, if you attach a wooden handle to the rod and then hold it by that handle as you rub the rod, the rod will remain charged and attract the piece of paper. 

The explanation for this is as follows: 

Without the insulating wood, the electric charges produced by rubbing readily move from the copper through your body and into the Earth. 

The insulating wooden handle prevents the flow of charge into your hand.

Semiconductors are a third class of materials, and their electrical properties are somewhere between those of insulators and those of conductors. Silicon and germanium are examples of semiconductors commonly used in the fabrication of a variety of electronic devices, such as transistors and light-emitting diodes. The electrical properties of semiconductors can be changed over many orders of magnitude by the addition of controlled amounts of certain atoms to the materials.

When a conductor is connected to the Earth by means of a conducting wire or pipe, it is said to be grounded. 

The Earth can then be considered an infinite “sink” or “Reservoir” to which electric charges can easily migrate. With this in mind, we can understand how to charge a conductor by a process known as induction.

To understand induction, consider a neutral (uncharged) conducting sphere insulated from ground, as shown in Figure 23.3a. e charge during this process.

When a negatively charged rubber rod is brought near the sphere, the region of the sphere nearest the rod obtains an excess of positive charge while the region farthest from the rod obtains an equal excess of negative charge, as shown in Figure 23.3b.

(That is, electrons in the region nearest the rod migrate to the opposite side of the sphere. This occurs even if the rod never actually touches the sphere.) because of the repulsive forces among the like charges.

If the same experiment is performed with a conducting wire connected from the sphere to ground (Fig. 23.3c),some of the electrons in the conductor are so strongly repelled by the presence of the negative charge in the rod that they move out of the sphere through theground wire and into the Earth.

If the wire to ground is then removed (Fig. 23.3d), the conducting sphere contains an excess of induced positive charge. When the rubber rod is removed from the vicinity of the sphere (Fig. 23.3e), this induced positive charge remains on the ungrounded sphere.Note that the charge remaining on the sphere is uniformly distributed over its surface because of the repulsive forces among the like charges. Also note that the rubber rod loses none of its negative charge during this process.

COULOMB’S LAWCoulomb’s experiments showed that the electric force between two stationary charged particles

• is inversely proportional to the square of the separation r between the particles and directed along the line joining them;

• is proportional to the product of the charges q1 and q2 on the two particles;

• is attractive if the charges are of opposite sign and repulsive if the charges have the same sign.

where is a constant called the Coulomb constant.

The smallest unit of charge known in nature is the Charge on an electron or proton,1 which has an absolute value of

From these observations, we can express Coulomb’s law as an equation giving the magnitude of the electric force (sometimes called the Coulomb force) between two point charges:

When dealing with Coulomb’s law, you must remember that force is a vector quantity and must be treated accordingly. Thus, the law expressed in vector form for the electric force exerted by a charge q1 on a second charge q2 , written F12 , is

where is a unit vector directed from q1 to q2

We define the strength (in other words, the magnitude) of the electric field at the location of the test charge to be the electric force per unit charge, or to be more specific the electric field E at a point in space is defined as

 The electric force Fe acting on a positive test charge q0 placed at that point divided by the magnitude of the test charge:              

Coulomb’s law is an inverse square law. THE ELECTRIC FIELDField forces can act through space, producing an effect even when no physical contact between the objects occurs.

Note that E is the field produced by some charge external to the test charge—it is not the field produced by the test charge itself. Also, note that the existence of an electric field is a property of its source. For example, every electron comes with its own electric field.

The gravitational field g at a point in space is defined to be equal to the gravitational force Fg  acting on a test particle of mass m divided by that mass: A similar approach to electric forces was developed by Michael Faraday.

The vector E has the SI units of newtons per coulomb (N/C).   Its direction is the direction of the force a positive test charge experiences when placed in the field. We say that an electric field exists at a point if a test charge at rest at that point experiences an electric force. Once the magnitude and direction of the electric field are known at some point, the electric force exerted on any charged particle placed at that point can be calculated from equation 23.3

Furthermore, the electric field is said to exist at some point (even empty space) regardless of whether a test charge is located at that point.(This is analogous to the gravitational field set up by any object, which is said to exist at a given point regardless of whether some other object is present at that point to “feel” the field.)

The electric field is directed radially outward from a positive charge and radially inward toward a negative charge.The electric field due to a group of point charges can be obtained by using the superposition principle. That is, the total electric field at some point equals the vector sum of the electric fields of all the charges:

Electric Field due to two charges

The electric force exerted on a charge of 2 X 10-8 C located at point P is qE = 5.4 X 10-3 N

The Electric dipole

An electric dipole is defined as a positive charge q and a negative charge q separated by some distance. For the dipole shown below, find the electric field E at P due to the charges, where P is a distance y >> a from the origin.

Thus, we see that, at distances far from a dipole but along the perpendicular bisector of the line joining the two charges, the magnitude of the electric field created by the dipole varies as 1/r3, whereas the more slowly varying field of a point charge varies as 1/r2 (see Eq. 23.4). This is because at distant points, the fields of the two charges of equal magnitude and opposite sign almost cancel each other.

Assignment:The Electric Field of a Uniform Ring of Charge A ring of radius a carries a uniformly distributed positive total charge Q. Calculate the electric field due to the ring at a point P lying a distance x from its center along the central axis perpendicular to the plane of the ring

ELECTRIC FIELD OF A CONTINUOUS CHARGE DISTRIBUTIONVery often the distances between charges in a group of charges are much smaller than the distance from the group to some point of interest (for example, a point where the electric field is to be calculated). In the system of charges that is smeared out, or continuous. That is, the system of closely spaced charges is equivalent to a total charge that is continuously distributed along some line, over some surface, or throughout some volume.The electric field at some point of a continuous charge distribution is

where dq is the charge on one element of the charge distribution and r is the distance from the element to the point in question.

ELECTRIC FIELD LINESA convenient way of visualizing electric field patterns is to draw lines that follow the same direction as the electric field vector at any point. Electric field lines describe an electric field in any region of space. The number of lines per unit area through a surface perpendicular to the lines is proportional to the magnitude of E in that region. These lines, called electric field lines, are related to the 

electric field in any region of space in the following manner:• The electric field vector E is tangent to the electric field line at each point.• The number of lines per unit area through a surface perpendicular to the lines is proportional to the magnitude of the electric field in that region. Thus, E is great when the field lines are close together and small when they are far apart.

Electric field lines penetrating two surfaces. The magnitude of the field is greater on surface A than on surface B.

Motion of Charged particle in a uniform electric field

Gauss’s LawAt beginning of this course we discussed how to use Coulomb’s law to calculate the electric field generated by a given charge distribution. In this section, we describe Gauss’s law and an alternative procedure for calculating electric fields.

The law is based on the fact that the fundamental electrostatic force between point charges exhibits an inverse-square behavior. 

Although a consequence of Coulomb’s law, Gauss’s law is more convenient for calculating the electric fields of highly symmetric charge distributions and makes possible useful qualitative reasoning when we are dealing with complicated problems.

ELECTRIC FLUXConsider an electric field that is uniform in both magnitude and direction, as shown in Figure 24.1. The field lines penetrate a rectangular surface of area A, which is perpendicular to the field. Recall from that the number of lines per unit area (in other words, the line density) is proportional to the magnitude of the electric field. 

Therefore, the total number of lines penetrating the surface is proportional to the product EA. This product of the magnitude of the electric field E and surface area A perpendicular to the field is called the electric flux E (uppercase Greek phi)

From the SI units of E and A, we see that E has units of newton–meters squared per coulomb (N.m2/C).  Electric flux is proportional to the number of electric field lines penetrating some surface.

Field lines representing a uniform electric field penetrating a plane of area A perpendicular to the field. The electric flux ФE  through this area is equal to EA.

Flux Through a Sphere

The field points radially outward and is therefore everywhere perpendicular to the surface of the sphere. The flux through the sphere is thus:

If the surface under consideration is not perpendicular to the field, the flux through it must be less than that given by Equation 24.1. We can understand this by considering Figure 24.2, in which the normal to the surface of area A is at an angle  to the uniform electric field.

 Note that the number of lines that cross this area A is equal to the number that cross the area A΄, which is a projection of area A aligned perpendicular to the field. From the Figure shown below we see that the two areas are related by A΄ = A cos θ . Because the flux through A equals the flux through A΄, we conclude that the flux through A is

In general, the electric flux through a surface is

You need to be able to apply Equations 24.2 and 24.3 in a variety of situations, particularly those in which symmetry simplifies the calculation.

Equation 24.3 is a surface integral, which means it must be evaluated over the surface in question. In general, the value of  ФE  depends both on the field pattern and on the surface.

where En represents the component of the electric field normal to the surface.

Figure 24.4 A closed surfaceA closed surface in an electric field. The area vectors Ai are, by convention, normal to the surface and point outward. The flux through an area element can be positive (element(1)), zero (element (2)), or negative (element (3)).

Flux Through a CubeThe net electric flux through the surface of a cube of edges, placed in a uniform electric field, oriented as shown in Figure below

The net flux is the sum of the fluxes through all faces of the cube. First, note that the flux through four of the faces (3), (4) , and the unnumbered ones) is zero because E is perpendicular to dA on these faces. The net flux through faces  (1) and (2)  is

Gauss’s law says that the net electric flux ФE  through any closed gaussian surface is equal to the net charge inside the surface divided by ϵ0 :

The net electric flux through a closed surface that surrounds no charge is zero.

APPLICATION OF GAUSS’S LAW TO CHARGED INSULATORS

The goal in this type of calculation is to determine a surface that satisfies one or more of the following conditions:1. The value of the electric field can be argued by symmetry to be constant over the surface.2. The dot product in Equation 24.6 can be expressed as a simple algebraic product E.dA because E and dA are parallel.3. The dot product in Equation 24.6 is zero because E and dA are perpendicular.4. The field can be argued to be zero over the surface.

The Electric Field Due to a Point Charge

Starting with Gauss’s law, calculate the electric field due to an isolated point charge q.

We choose a spherical gaussian surface of radius r centered on the point charge, as shown in Figure above. The electric field due to a positive point charge is directed radially outward by symmetry and is therefore normal to the surface at every point. Thus, as in condition (2), E is parallel to dA at each point. Therefore, E.dA = EdA and Gauss’s law gives

By symmetry, E is constant everywhere on the surface, which satisfies condition (1), so it can be removed from the integral.Therefore,

where we have used the fact that the surface area of a sphere is 4πr2. Now, we solve for the electric field:

An insulating solid sphere of radius a has a uniform volume charge density, ρ and carries a total positive charge Q (Fig.  a)   (a) Calculate the magnitude of the electric field at a point outside the sphere. (b) Find the magnitude of the electric field at a point inside the sphere.

Because the charge distribution is spherically symmetric, we again select a spherical gaussian surface of radius r, concentric with the sphere, as shown in Figure a. For this choice, conditions (1) and (2) are satisfied, as they were for the point charge and we find that

Because the charge distribution is spherically symmetric, we again select a spherical gaussian surface of radius r, concentric with the sphere, as shown in Figure a. For this choice, conditions (1) and (2) are satisfied, as they were for the point charge and we find that

Note that this result is identical to the one we obtained for a point charge. Therefore, we conclude that, for a uniformlycharged sphere, the field in the region external to the sphere is equivalent to that of a point charge located at the center of the sphere.

In this case we select a spherical gaussian surface having radius r a, concentric with the insulated sphere (Fig. b). Let us denote the volume of this smaller sphere by V .

To apply Gauss’s law in this situation, it is important to recognize that the charge qin within the gaussian surface of volume V is less than Q. To calculate qin , we use the fact that

Therefore, Gauss’s law in the region gives

Note that this result for E differs from the one we obtained in part (a). It shows that E → 0 as r → 0. Therefore, the result eliminates the problem that would exist at r = 0 if E varied as 1/r2 inside the sphere as it does outside the sphere.

Typical Electric Field Calculations Using Gauss’s Law

A conductor in electrostatic equilibrium has the following properties:

1. The electric field is zero everywhere inside the conductor.2. Any net charge on the conductor resides entirely on its surface.3. The electric field just outside the conductor is perpendicular to its surface and has a magnitude σ/ϵ0 , where σ is the surface charge density at that point.4. On an irregularly shaped conductor, the surface charge density is greatest where the radius of curvature of the surface is the smallest.

Assignment:

A Cylindrically Symmetric Charge DistributionFind the electric field a distance r from a line of positive charge of infinite length and constant charge per unit length, λ.

Since the electrostatic force given by Coulomb’s law is conservative,electrostatic phenomena can be conveniently described in terms of an electric potential energy. 

This idea enables us to define a scalar quantity known as electric potential.

Because the electric potential at any point in an electric field is a scalar function, we can use it to describe electrostatic phenomena more simply than if we were to rely only on the concepts of the electric field and electric forces.

POTENTIAL DIFFERENCE AND ELECTRIC POTENTIAL

The force  q0E is conservative because the individual forces described by Coulomb’s law are conservative. When the test charge is moved in the field by some external agent, the work done by the field on the charge is equal to the negative of the work done by the external agent causing the displacement.

 For an infinitesimal displacement ds, the work done by the electric field on the charge is F.dS = q0E.dS. As this amount of work is done by the field, the potential energy of the charge–field system is decreased by an amount  

dU = -q0E.dS . For a finite displacement of the charge from a point A to a point B, the change in potential energy of the system ΔU = UB – UA is 

Because the force q0E is conservative, this line integral does not depend on the path taken from A to B.The potential energy per unit charge U/q0 is independent of the value of q0 and has a unique value at every point in an electric field. This quantity U/q0 is called the electric potential (or simply the potential) V. Thus, the electric potential at any point in an electric field is  

The fact that potential energy is a scalar quantity means that electric potential also is a scalar quantity and has units of Joules per coulomb (J/C). The potential difference V between points A and B in an electric field E is defined as

Electric potential at an arbitrary point in an electric field equals the work required per unit charge to bring a positive test charge from infinity to that point. Thus, if we take point A in Equation 25.3 to be at infinity, the electric potential at any point P is

SI unit of both electric potential and potential difference is joules per coulomb, which is defined as a volt (V):

POTENTIAL DIFFERENCES IN A UNIFORM ELECTRIC FIELDEquations 25.1 and 25.3 hold in all electric fields, whether uniform or varying, but they can be simplified for a uniform field. First, consider a uniform electric field directed along the negative y axis, as shown in Figure 25.1a. Let us calculate the potential difference between two points A and B separated by a distance d, where d is measured parallel to the field lines. Equation 25.3 gives

From this result, we see that if q0 is positive, then ΔU is negative. We conclude that a positive charge loses electric potential energy when it moves in the direction of the electric field. This means that an electric field does work on a positive charge when the charge moves in the direction of the electric field.

From this result, we see that if q0 is positive, then ΔU is negative. We conclude that a positive charge loses electric potential energy when it moves in the direction of the electric field. This means that an electric field does work on a positive charge when the charge moves in the direction of the electric field. (This is analogous to the work done by the gravitational field on a falling mass, as shown in Figure 25.1b.) If a positive test charge is released from rest in this electric field, it experiences an electric force q0E in the direction of E downward in Fig. 25.1a).Therefore, it accelerates downward, gaining kinetic energy. As the charged particle gains kinetic energy, it loses an equal amount of potential energy.

If q0 is negative, then ΔU is positive and the situation is reversed: A negative charge gains electric potential energy when it moves in the direction of the electric field. If a negative charge is released from rest in the field E, it acceleratesin a direction opposite the direction of the field.

Now consider the more general case of a charged particle that is free to movebetween any two points in a uniform electric field directed along the x axis, asshown in Figure 25.2.

Figure 25.2 A uniform electricfield directed along the positive x axis. Point B is at a lower electric potential than point A. Points B and C are at the same electric potential.

If s represents the displacement vector between points A and B, Equation 25.3 give

ELECTRIC POTENTIAL AND POTENTIAL ENERGY DUE TO POINT CHARGES

When we consider an isolated positive point charge q, it should be noted that such a charge produces an electric field that is directed radially outward from the charge. To find the electric potential at a point located a distance r from the charge, we begin with the general expression for potential difference:

where A and B are the two arbitrary points shown in Figure below

The potential difference between points A and B due to a point charge q depends only on the initial and final radial coordinates rA  and  rB. The two dashed circles represent cross-sections of spherical equipotential surfaces.

The E.dS integral of is independent of the path between points A and B—as it must be because the electric field of a point charge is conservative.

Furthermore, Equation 25.10 expresses the important result that the potential difference between any two points A and B in a field created by a point charge depends only on the radial coordinates rA  and  rB . It is customary to choose the reference of electric potential to be zero at With this reference, the electric potential created by a point charge at any distance r from the charge is

The total electric potential at some point P due to several point charges is the sum of the potentials due to the individual charges. For a group of point charges, we can write the total electric potential at P in the form

where the potential is again taken to be zero at infinity and ri  is the distance from the point P to the charge qi.

The potential energy associated with a pair of point charges separated by a distance r12  is

If we know the electric potential as a function of coordinates x, y, z, we can obtain the components of the electric field by taking the negative derivative of the electric potential with respect to the coordinates. For example, the x component of the electric field is

Every point on the surface of a charged conductor in electrostatic equilibrium is at the same electric potential. The potential is constant everywhere inside the conductor and equal to its value at the surface.

CAPACITORSA capacitor consists of two conductors separated by an insulator. We shall see that the capacitance of a given capacitor depends on its geometry and on the material— called a dielectric—that separates the conductors.

DEFINITION OF CAPACITANCE

CALCULATING CAPACITANCE

We can calculate the capacitance of a pair of oppositely charged conductors in the following manner: We assume a charge of magnitude Q , and we calculate the potential difference. We then use the expression to evaluate the capacitance C = Q/ΔV. As we might expect, we can perform this calculation relatively easily if the geometry of the capacitor is simple.

We can calculate the capacitance of an isolated spherical conductor of radius R and charge Q if we assume that the second conductor making up the capacitor is a concentric hollow sphere of infinite radius. The electric potential of the sphere of radius R is simply of radius R is simply keQ /R, and setting at infinity as usual, we have ke Q /R, and setting V = 0 at infinity as usual, we have

Parallel-Plate CapacitorsTwo parallel metallic plates of equal area A are separated by a distance d, as shown in Figure 26.2. One plate carries a charge Q , and the other carries a charge Q.

the value of the electric field between the plates is

That is, the capacitance of a parallel-plate capacitor is proportional to the area of its plates and inversely proportional to the plate separation, just as we expect from our conceptual argument.

Assignment:Calculate the capacitance of the other familiar geometries: cylinders and spheres.

COMBINATIONS OF CAPACITORS

Two or more capacitors often are combined in electric circuits. We can calculate the equivalent capacitance of certain combinations using methods described in this section.

Parallel Combination

Series Combination

ENERGY STORED IN A CHARGED CAPACITOR

The work necessary to transfer an increment of charge dq from the plate carrying charge q to the plate carrying charge q (which is at the higher electric potential) is

This result applies to any capacitor, regardless of its geometry. We see that for a given capacitance, the stored energy increases as the charge increases and as the potential difference increases. In practice, there is a limit to the maximum energy (or charge) that can be stored because, at a sufficiently great value of V, discharge ultimately occurs between the plates. For this reason, capacitors are usually labeled with a maximum operating voltage.

The energy density in any electric field is proportional to the square of the magnitude of the electric field at a given point.

CAPACITORS WITH DIELECTRICSA dielectric is a nonconducting material, such as rubber, glass, or waxed paper. When a dielectric is inserted between the plates of a capacitor, the capacitance increases.If the dielectric completely fills the space between the plates, the capacitance increases by a dimensionless factor , which is called the dielectric constant. The dielectric constant is a property of a material and varies from one material to another.

For a parallel-plate capacitor, where (Eq. 26.3), we can express the capacitance when the capacitor is filled with a dielectric as